{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Some Variations of Banach's Matchbox Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Banach's matchbox problem is a good entry point into stochastic stopping problems. A man buys two matchbooks and puts one in each of his two pockets. He then selects a matchbox at random from either pocket, uses a single match, and then returns the matchbox to the same pocket. This problem is slightly different from the classic matchbox problem in that when the last match is taken, the sequence ends. In the classic problem, discovery of the empty box happens only when the empty box is selected **after** being emptied.\n",
    "\n",
    "The question is: **What is the probability of $k$ matches in the remaining matchbox? **\n",
    "\n",
    "The problem is given two matchbooks containing $n$ matches each and with a matchbook placed in both the left and right pockets, if a person reaches into one or the other pocket at random with equal probability, what is the probability of there being $k$ matches in the other pocket when the matchbook in the selected pocket is found to be empty?  That is, the person keeps reaching alternatively and at random to sample a match from either pocket until one of the pockets is exhausted of matches.\n",
    "\n",
    "This is easy to code up in Python using a generator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import  division\n",
    "from collections import Counter, OrderedDict\n",
    "import pandas as pd\n",
    "import random\n",
    "import numpy as np\n",
    "from scipy.misc import comb\n",
    "random.seed(12345)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def step(n=4):\n",
    "  'keep track of remaining matches in each matchbook'\n",
    "  a = b = n\n",
    "  while a>0 and b>0:\n",
    "    if random.randint(0,1):\n",
    "        a-=1\n",
    "    else:\n",
    "        b-=1\n",
    "    yield (a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, suppose there are $n=4$ matches in each matchbook, then a valid sequence of draws from the `(left,right`) pocket is the following:\n",
    "\n",
    "`[(4, 3), (4, 2), (3, 2), (3, 1), (3, 0)]`\n",
    "\n",
    "This means that the first draw is from the right pocket leaving `3` matches there and `4` matches in the left pocket. The next draw again\n",
    "samples a match from the right pocket leaving `2` matches there and `4` in the left pocket. For the following draw, the left pocket is chosen leaving `3` matches there and `2` matches in the right pocket. This continues until the right pocket is emptied `(3,0)`. We can draw this sequence using the following code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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3Nj0RAPT39zM6Olqu7g4PD/ueCCCYcfkRUzQarTv1ejweZ2Zmpnxq0NXVRTQarVm+VCrV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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e13828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from __future__ import division\n",
    "%matplotlib inline\n",
    "from matplotlib.pylab import subplots,mgrid\n",
    "\n",
    "def draw_grid(n=4):\n",
    "    'draw square grid of `n` dimensions'\n",
    "    fig,ax = subplots()\n",
    "    i,j=mgrid[0:n+1,0:n+1]\n",
    "    ax.scatter(i.flat,j.flat,alpha=.8,color='black')\n",
    "    ax.set_aspect(1)\n",
    "    ax.plot(0,0,'ow',mec='w'); # remove origin\n",
    "    ax.plot([0,]*4,range(1,5),'or',mec='r',ms=10)\n",
    "    ax.plot(range(1,5),[0,]*4,'or',mec='r',ms=10)\n",
    "    return ax\n",
    "\n",
    "def draw_path(seq,ax,color='gray',alpha=0.5):\n",
    "    x,y=zip(*seq)\n",
    "    n = max(seq[0])\n",
    "    ax.plot((n,)+x,(n,)+y,marker='o',markersize=20,\n",
    "            alpha=alpha,color=color,lw=5)\n",
    "    ax.set_title('Sequence Length=%d'%(len(x)))    \n",
    "            \n",
    "ax = draw_grid()\n",
    "draw_path([(4, 3), (4, 2), (3, 2), (3, 1), (3, 0)],ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the figure above, the red circles indicate the termination points where one of the pockets has been emptied. The `(4,4)` point is the starting point with incremental steps moving down and to the left until one of the red circles is encountered. The length of the sequence is indicated in the title. In this case it took five draws in total to exhaust one of the matchbooks and terminate the sequence.\n",
    "\n",
    "The classical matchbox problem is to find the probability of termination at a particular circle. For example, what is the probability that the sequence terminates with one match remaining in the other matchbook? In the figure above, this means terminating at `(1,0)` or `(0,1)`.\n",
    "\n",
    "Specifically, termination at `(1,0)` means accumulating four steps down and three steps left in any sequence. This is the same as the $n$ *choose* $k$ binomial coefficient $\\texttt{Binom}(n,k)$. We can compute this using `scipy` as the following with $n=7,k=3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "35\n"
     ]
    }
   ],
   "source": [
    "print comb(7,3,exact=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The problem with this approach is that we can accidentally count paths that would have terminated earlier. For example,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ZjgYGBoquaWwsycWLQyQSwpkzIWDKABIEgynKy5XGxioikaVlUy42lunIyJIr32AuMpkM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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dfc2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = draw_grid()\n",
    "draw_path([(3, 4), (3, 3), (2, 3), (2, 2), (1, 2), (1, 1), (1, 0)],ax)\n",
    "draw_path([(4,3), (4, 2), (4, 1), (4, 0), (3, 0), (2, 0), (1, 0)],ax,'blue',.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The blue path would never have gotten so long because it would have encountered the termination point at `(4,0)`. Thus, this straight-forward counting scheme would over-count by including these paths. The following figure shows these valid paths that terminate at `(1,0)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4cWPS1G/zcblcbNy4kTNnziTNI5lpbk4zmJiAc+esLYUMOcvl2Wefpa6ublk/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\nwTYTdyeVVacAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x198b9cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "paths=[[(3, 4),(2, 4),(1, 4),(1, 3),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(2, 4),(2, 3),(1, 3),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(2, 4),(2, 3),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(2, 4),(2, 3),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(2, 3),(1, 3),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(2, 3),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(2, 3),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(3, 2),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(3, 2),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(3, 4),(3, 3),(3, 2),(3, 1),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(2, 3),(1, 3),(1, 2),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(2, 3),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(2, 3),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(3, 2),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(3, 2),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(3, 3),(3, 2),(3, 1),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(4, 2),(3, 2),(2, 2),(1, 2),(1, 1),(1, 0)],\n",
    "[(4, 3),(4, 2),(3, 2),(2, 2),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(4, 2),(3, 2),(3, 1),(2, 1),(1, 1),(1, 0)],\n",
    "[(4, 3),(4, 2),(4, 1),(3, 1),(2, 1),(1, 1),(1, 0)]]\n",
    "\n",
    "ax = draw_grid()\n",
    "for i in paths:\n",
    "    draw_path(i,ax,alpha=.1)\n",
    "   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's change our perspective slightly. Instead let's examine the probability of a sequence of a certain length and see if we can use that to answer the classic question. To start with, the following figure shows valid four-long sequences. The diagonal elements are indicated in green and these are the termination points for all four-long sequences. There are $2^4=16$ such sequences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9scdg3TqoqnKmdeuceVYIzCza2to86SwUj8d9P7oqZnl1R66vd/oRjIzAm286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\nBoD/Dy+l+hm56tpSAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x19899ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = draw_grid()\n",
    "for i in zip(range(5)[::-1],range(5)):\n",
    "    ax.plot(i[0],i[1],'sg',ms=15,alpha=.3)\n",
    "draw_path([(3,4),(2,4),(1,4),(0,4)],ax)\n",
    "draw_path([(4,3),(4,2),(4,1),(4,0)],ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because only two of the sixteen valid  sequences result in termination, the probability of termination with a four-long sequence is $P_4 = \\frac{2}{16}= \\frac{1}{8}$. Now let's examine the diagonal elements with the following labels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1988feb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = draw_grid()\n",
    "for i in zip(range(1,4)[::-1],range(1,4)):\n",
    "    ax.text(i[0],i[1],'(%d,%d)'%i,fontsize=13,color='g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the labels to compute the number of paths that terminate at each of the diagonal elements.\n",
    "\n",
    "| Label | Number of Paths|\n",
    "|-------|----------------|\n",
    "| (1,3) | $\\texttt{binom}(4,1)=4$|\n",
    "| (2,2) | $\\texttt{binom}(4,2)=6$|\n",
    "| (3,1) | $\\texttt{binom}(4,1)=4$|\n",
    "\n",
    "This means that there are `16-2=4+6+4=14` paths out of the initial group of `16` that have yet to terminate. Now, let's consider sequences of length five. The following figure shows the migration of paths to the next lower diagonal corresponding to the five-long sequences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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VOAi82DoIvKTpGMH5BxAjHgRWIXDVVV23mzkzsm8kL9Y1IB4mTOi63cRJzUEQ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\nBkopm4aBUgrQMFBK2TQMlFIA/H80UN5iu1jasQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a8b5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# construct matrix of indices for convenience\n",
    "i,j=np.meshgrid(range(5),range(4,-1,-1))\n",
    "m= i+1j*j\n",
    "\n",
    "ax = draw_grid()\n",
    "for i in np.diagonal(m,-1):\n",
    "    ax.plot(i.real,i.imag,'sg',ms=15,alpha=.3)\n",
    "\n",
    "for i in np.diagonal(m,0):\n",
    "    if i.real==0 or i.imag==0: continue\n",
    "    ax.text(i.real,i.imag,'(%d,%d)'%(i.real,i.imag),fontsize=13,color='g')\n",
    "\n",
    "# add arrow markers\n",
    "heads=map(lambda i:(i.real,i.imag),np.diagonal(m,0))[1:-1]\n",
    "\n",
    "for i,j in heads:\n",
    "    t=(i-1,j)\n",
    "    ax.annotate(\"\",xy=(i-1,j),xytext=(i,j),\n",
    "                arrowprops=dict(width=.5,color='green'),\n",
    "                )\n",
    "    ax.annotate(\"\",xy=(i,j-1),xytext=(i,j),\n",
    "                arrowprops=dict(width=.5,color='green'),\n",
    "                )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that two arrows end up at red termini. There are `28=2*14` paths in total (because each diagonal element can go either down or right). Of these `28`, eight terminate on a red circle. Thus, under these conditions, the probability of a terminating five-long sequence is therefore $p_5 = \\frac{8}{28}=\\frac{2}{7}$. Next, we can draw the next layer corresponding to six-long sequences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WwLe/DVdfDQcP2qMA4rlwMy/PblrU9RReftm22bDB7muIjLSbFM8+27COtWthyhSI18GA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\nAPj/a1z2S73q3q4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a82320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = draw_grid()\n",
    "for i in np.diagonal(m,-2):\n",
    "#     if i.real==0 or i.imag==0: continue\n",
    "    ax.plot(i.real,i.imag,'sg',ms=15,alpha=.3)\n",
    "    \n",
    "for i in np.diagonal(m,-1):\n",
    "    if i.real==0 or i.imag==0: continue\n",
    "    ax.text(i.real,i.imag,'(%d,%d)'%(i.real,i.imag),fontsize=13,color='g')\n",
    "\n",
    "for i in np.diagonal(m,-2):\n",
    "#     if i.real==0 or i.imag==0: continue\n",
    "    ax.text(i.real-0.5,i.imag-0.5,'(%d,%d)'%(i.real,i.imag),fontsize=13,color='blue')\n",
    "\n",
    "    \n",
    "# add arrow markers\n",
    "heads=map(lambda i:(i.real,i.imag),np.diagonal(m,-1))[1:-1]\n",
    "\n",
    "for i,j in heads:\n",
    "    t=(i-1,j)\n",
    "    ax.annotate(\"\",xy=(i-1,j),xytext=(i,j),\n",
    "                arrowprops=dict(width=.5,color='green'),\n",
    "                )\n",
    "    ax.annotate(\"\",xy=(i,j-1),xytext=(i,j),\n",
    "                arrowprops=dict(width=.5,color='green'),\n",
    "                )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the last stage, there were `8` out of `28` paths that terminated. That leaves `20` paths still in play and therefore `40` paths on the indicated diagonal.  In terms of our earlier accounting, we have\n",
    "\n",
    "| Label | Number of Paths|\n",
    "|-------|----------------|\n",
    "| <span style=color:green>(1,2)</span> | 10|\n",
    "| <span style=color:green>(2,1)</span> | 10|\n",
    "\n",
    "\n",
    "For the next layer, we have the following:\n",
    "\n",
    "| Label | Number of Paths|\n",
    "|-------|----------------|\n",
    "| <span style=color:blue>(0,2)</span> | 10|\n",
    "| <span style=color:blue>(2,0)</span> | 10|\n",
    "| <span style=color:blue>(1,1)</span>  | 20|\n",
    "\n",
    "With this accounting, the probability of termination with a six-long\n",
    "sequence is $p_6=\\frac{20}{40}=\\frac{1}{2}$. Finally, because all paths going through `(1,1)` terminate at either `(1,0)` or `(0,1)`, we have $p_7=1$.\n",
    "\n",
    "To assemble all these results, recall we have\n",
    "$$ P_4=\\frac{1}{8}$$\n",
    "\n",
    "and also, because $p_4 = P(T_5| \\hat{T}_4)$ where $T_5$ is the probability of termination in five steps and $\\hat{T}_4$ is the probability of not terminating in four steps, we have\n",
    "\n",
    "$$ P_5 = p_5 (1-P_4) = \\frac{2}{7} \\frac{7}{8}=\\frac{1}{4}$$\n",
    "\n",
    "Likewise for $P_6$,\n",
    "\n",
    "$$ P_6 = p_6 (1-P_4)(1-p_5) = \\frac{1}{2}\\frac{7}{8}\\frac{5}{7} =\\frac{5}{16}$$\n",
    "\n",
    "Finally, for $P_7$, we have the following:\n",
    "\n",
    "$$ P_7 = p_7 (1-P_4)(1-p_5)(1-p_6) = 1\\frac{7}{8}\\frac{5}{7}\\frac{1}{2} =\\frac{5}{16}$$\n",
    "\n",
    "Note that\n",
    "\n",
    "$$ P_4+P_5+P_6+P_7 = 1 $$\n",
    "\n",
    "A quick simulation can bear this out. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({4: 125, 5: 232, 6: 330, 7: 313})"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Counter([len(tuple(step())) for i in range(1000)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "We can automate this process in the following code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.misc import comb\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 4 6 4 1]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(4, 4)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from __future__ import division\n",
    "n=4\n",
    "t = np.array([comb(n,i,exact=True) for i in range(n+1)],dtype=int)\n",
    "print t\n",
    "t[[0,-1]].sum()/t.sum()\n",
    "# split\n",
    "(t[1],t[-2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4, 10, 10,  4])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.lib.stride_tricks import as_strided\n",
    "np.hstack([t[1],as_strided(t[1:-1],(2,3-2+1),(t.itemsize,)*2).sum(axis=1),t[-2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.misc import comb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 4, 6, 4, 1]\n",
      "0.125\n"
     ]
    }
   ],
   "source": [
    "x=[comb(4,i,exact=True) for i in range(5)]\n",
    "print x\n",
    "print x[0]*2/sum(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a = x.pop(0)\n",
    "b = x.pop()\n",
    "x=[ x[0] ]+[ sum(x[slice(i,i+2)]) for i in range(len(x)-1) ]+[ x[-1] ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.285714285714\n"
     ]
    }
   ],
   "source": [
    "print 2*x[0]/sum(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def probstop(n=4):\n",
    "    o=OrderedDict()\n",
    "    k=n\n",
    "    x=[comb(n,i,exact=True) for i in range(n+1)]\n",
    "    o[k]=2*x[0]/sum(x)\n",
    "    while len(x)>2:\n",
    "        a = x.pop(0)\n",
    "        b = x.pop()\n",
    "        x=[x[0]]+[sum(x[slice(i,i+2)]) for i in range(len(x)-1)]+[x[-1]]\n",
    "        k+=1\n",
    "        o[k]=2*x[0]/sum(x)\n",
    "        assert o[k]>0\n",
    "    p = OrderedDict()\n",
    "    factor=1\n",
    "    for k,v in o.iteritems():\n",
    "        p[k]=factor*o[k]\n",
    "        factor = factor*(1-v)        \n",
    "    return o,p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "o,p=probstop(30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedDict([(30, 1.862645149230957e-09),\n",
       "             (31, 2.7939677238464355e-08),\n",
       "             (32, 2.1653249859809875e-07),\n",
       "             (33, 1.1548399925231934e-06),\n",
       "             (34, 4.763714969158173e-06),\n",
       "             (35, 1.6196630895137787e-05),\n",
       "             (36, 4.724017344415188e-05),\n",
       "             (37, 0.0001214747317135334),\n",
       "             (38, 0.000280910317087546),\n",
       "             (39, 0.0005930328916292638),\n",
       "             (40, 0.0011564141386770643),\n",
       "             (41, 0.002102571161231026),\n",
       "             (42, 0.003591892400436336),\n",
       "             (43, 0.005802287723781774),\n",
       "             (44, 0.008910656147236296),\n",
       "             (45, 0.0130689623492799),\n",
       "             (46, 0.01837822830367486),\n",
       "             (47, 0.024864661822618928),\n",
       "             (48, 0.032462197379530267),\n",
       "             (49, 0.041004880900459284),\n",
       "             (50, 0.05023097910306262),\n",
       "             (51, 0.059798784646503116),\n",
       "             (52, 0.0693122276584468),\n",
       "             (53, 0.07835295300520073),\n",
       "             (54, 0.08651471894324247),\n",
       "             (55, 0.09343589645870187),\n",
       "             (56, 0.0988264289467039),\n",
       "             (57, 0.10248666705584109),\n",
       "             (58, 0.10431678611040969),\n",
       "             (59, 0.10431678611040969)])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(p.values())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a978cc0>]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HqIhIwSxYEFTQnDYt7EikkNTTF5E9vP128BSshx6CkSPDjkay0fCOiBTEeecF1TNnzQo7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\nRElfRKSKKOmLiFSR/w+DrDT5Lt0oAwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a4c0c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax=subplots()\n",
    "ax.plot(p.keys(),p.values(),'-o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1a986710>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w=pd.Series(Counter([len(tuple(step(30))) for i in range(5000)]))\n",
    "(w/5000.).plot(ax=ax,marker='s')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9PP00/O7VLMYv+jvpI1d7HKHEqmiS+2ag5CqKLuGysnW6RqpjZk0JJfYXnHOv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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a4c0c18>"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cBjyYt30F3Y7eyRbzr+RtrwD26OG5ej1CMxv40Vwh44di7IgRp0ceQ5j/bh8+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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x19f017b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax=subplots()\n",
    "ax.plot(p.keys(),np.cumsum(p.values()),'-o')\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 100.0, -1, 100.0)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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pPjKwkoxcXs/PA3YpbcPGR4TQZMS8AB6Vv1LmyTBDuHpCf5E+oZTSEuhkfYCQ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\nnEchsucUKgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1fede128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax=subplots()\n",
    "fig.set_size_inches((10,5))\n",
    "for i in range(150):\n",
    "    x,y=zip(*tuple(step(100)))\n",
    "    ax.plot(x,y,'o-',color='gray',alpha=.1/5.)\n",
    "    ax.plot(x[-1],y[-1],'sg',alpha=.3)\n",
    "ax.set_aspect(1)\n",
    "ax.axis(xmin=-1,ymin=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
